Parton distributions in the virtual photon up to NNLO in QCD and - - PowerPoint PPT Presentation

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Apr 01, 2023 145 likes •430 views

Parton distributions in the virtual photon up to NNLO in QCD and factorization scheme dependence Ken Sasaki (Yokohama National U.) In collaboration with T. Uematsu (Kyoto U.) T. Ueda (KEK) RADCOR 2007 at GGI, Florence Introduction &

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Parton distributions in the virtual photon up to NNLO in QCD and factorization scheme dependence Ken Sasaki (Yokohama National U.) In collaboration with

T. Uematsu (Kyoto U.)
T. Ueda (KEK)

RADCOR 2007 at GGI, Florence

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Introduction & Motivation

Deep-inelastic electron-nucleon scattering

Parton distribution functions (PDFs) inside a nucleon

Rather difficult to see the features of factorization schemes

factorization-scheme dependent

 necessary for the analysis of semi-inclusive

reactions

probe target DIS Nucleon structure functions

:Bjorken variable :mass squared of the probe photon Some assumptions made to extract PDFs from data

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Introduction & Motivation

Future linear collider experiment (e.g. ILC) Highly virtual photon target

Two-photon process Deep inelastic scattering target probe large Viewed as a deep-inelastic electron-photon scattering We can study the structures

f photon

:mass squared of the target photon

pQCD gives definite predictions for

pdf in the virtual photon
A good playground to see the scheme-dependence of pdf

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in Perturbative QCD

For highly virtual photon target ( )

(LO) (NLO)

Uematsu-Walsh (1981,1982)

: QCD scale parameter Hadronic piece can also be dealt with perturbatively Definite prediction of , its shape and magnitude, is possible

Motivated by the calculation of 3-loop anomalous dimensions Vogt-Moch-Vermaseren (2004,2006)

(NNLO) LO, NLO NNLO

Ueda-Uematsu-Sasaki

Physical Review D75, 114009 (2007) hep-ph/0703296

(Uematsu’ talk) (OPE+RG method)

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QCD-improved Parton Model Approach

Unpolarized PDFs In LO in QED coupling : Singlet quark DF: Non-singlet quark DF: :Quark DF with helicity of the virtual photon ( ) : Gluon DF with helicity : Photon DF with helicity

Photon DF does not evolve

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QCD-improved Parton Model Approach

Parton distribution functions in the virtual photon Factorization: DGLAP evolution equation :Splitting fn. of photon into quark and gluon :Splitting fn. of quark and gluon

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QCD-improved Parton Model Approach

Taking moments: Expanding in powers of Introducing

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QCD-improved Parton Model Approach

Solution to DGLAP evolution eq. Initial conditions (factorization scheme dependent) Splitting functions anomalous dimensions When parameters calculated in scheme are used, we obtain PDFs in the virtual photon up to NNLO in scheme finite photon matrix element

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QCD-improved Parton Model Approach

Solution

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Moment Sum Rule for

Since NNLO ( ) LO ( ) NLO ( )

for even

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Moment Sum Rule for

LO : NLO :

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Moment Sum Rule for NNLO :

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Moment Sum Rule for NNLO :

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Factorization-scheme independent combinations

Since is the moments of are physically measurable quantities,

each term, , is factorization-scheme (FS) independent

In the NS sector, the following combinations are FS independent

Bardeen-Buras (1979) Nucleon structure func

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PDFs in scheme

and

In the expression of , put

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PDFs in scheme

Non-singlet quark

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PDFs in scheme

Singlet quark

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PDFs in scheme

Gluon

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Factorization scheme dependence

Scheme-dependent PDFs

DIS scheme

scheme

Gluck-Reya-Vogt (1992)

Photonic coefficient function in NLO becomes negative and divergent for . It is included into quark PDFs.

Altarelli-Ellis-Martinelli(1978)

is given by the naïve parton model expression to all orders scheme independent

scheme

(hadronic coefficient functions are the same as the tree level)

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PDFs in DIS scheme

In DIS scheme, the coefficient fns are those at the tree level In the expression of , put

and

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PDFs in DIS scheme

Singlet quark

(LO) (NLO) (NNLO)

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PDFs in scheme

Expressing and as

Then we get

In scheme, photon coefficient fn is absorbed into quark distribution fns, so that

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PDFs in scheme

Singlet quark

(LO) (NLO) (NNLO)

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PDFs in scheme

for Singlet quark

(LO) (NLO) (NNLO)

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PDFs in scheme

Large behavior

Behavior at

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Summary

PDFs (quark and gluon) in the virtual photon were investigated up to NNLO ( ) in the kinematical region PDFs were studied in , DIS and schemes Scheme dependences appear at large

Singlet quark

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Parton distributions in the virtual photon up to NNLO in QCD and factorization scheme dependence Ken Sasaki (Yokohama National U.) In collaboration with

Introduction & Motivation

Deep-inelastic electron-nucleon scattering

Parton distribution functions (PDFs) inside a nucleon

probe target DIS Nucleon structure functions

Introduction & Motivation

Future linear collider experiment (e.g. ILC) Highly virtual photon target

Two-photon process Deep inelastic scattering target probe large Viewed as a deep-inelastic electron-photon scattering We can study the structures

pQCD gives definite predictions for

in Perturbative QCD

For highly virtual photon target ( )

(LO) (NLO)

Uematsu-Walsh (1981,1982)

: QCD scale parameter Hadronic piece can also be dealt with perturbatively Definite prediction of , its shape and magnitude, is possible

Motivated by the calculation of 3-loop anomalous dimensions Vogt-Moch-Vermaseren (2004,2006)

(NNLO) LO, NLO NNLO

Ueda-Uematsu-Sasaki

(Uematsu’ talk) (OPE+RG method)

QCD-improved Parton Model Approach

Unpolarized PDFs In LO in QED coupling : Singlet quark DF: Non-singlet quark DF: :Quark DF with helicity of the virtual photon ( ) : Gluon DF with helicity : Photon DF with helicity

QCD-improved Parton Model Approach

Parton distribution functions in the virtual photon Factorization: DGLAP evolution equation :Splitting fn. of photon into quark and gluon :Splitting fn. of quark and gluon

QCD-improved Parton Model Approach

Taking moments: Expanding in powers of Introducing

QCD-improved Parton Model Approach

Solution to DGLAP evolution eq. Initial conditions (factorization scheme dependent) Splitting functions anomalous dimensions When parameters calculated in scheme are used, we obtain PDFs in the virtual photon up to NNLO in scheme finite photon matrix element

QCD-improved Parton Model Approach

Solution

Moment Sum Rule for

Since NNLO ( ) LO ( ) NLO ( )

for even

Moment Sum Rule for

LO : NLO :

Moment Sum Rule for NNLO :

Moment Sum Rule for NNLO :

Factorization-scheme independent combinations

Since is the moments of are physically measurable quantities,

each term, , is factorization-scheme (FS) independent

In the NS sector, the following combinations are FS independent

PDFs in scheme

and

PDFs in scheme

Non-singlet quark

PDFs in scheme

Singlet quark

PDFs in scheme

Gluon

Factorization scheme dependence

Scheme-dependent PDFs

DIS scheme

scheme

Gluck-Reya-Vogt (1992)

Altarelli-Ellis-Martinelli(1978)

scheme

PDFs in DIS scheme

In DIS scheme, the coefficient fns are those at the tree level In the expression of , put

and

PDFs in DIS scheme

Singlet quark

PDFs in scheme

Expressing and as

Then we get

In scheme, photon coefficient fn is absorbed into quark distribution fns, so that

PDFs in scheme

Singlet quark

PDFs in scheme

for Singlet quark

PDFs in scheme

Large behavior

Behavior at

Summary

PDFs (quark and gluon) in the virtual photon were investigated up to NNLO ( ) in the kinematical region PDFs were studied in , DIS and schemes Scheme dependences appear at large

Summary

Gluon distribution